Fans of anime and manga know that it's extremely common for manga series to be adapted into anime, more so than original stories being created as anime directly. However, while some adaptations are fantastically faithful to the manga, such as Jojo's Bizarre Adventure, others... aren't. There are many reasons why this might be the case; in some situations, it's because the manga isn't finished being written yet, and the anime ran out of material to adapt. In others, it was never meant to be the exact same, but rather more "inspired" by the manga. In a few cases, it even happens because fans were unhappy with the manga, and the anime was written with those complaints in mind.
Because it's difficult to compare how different things are, these entries aren't ranked in any particular way. This list is primarily focused on how and why the anime diverged from the manga, and much less so whether or not it was a good thing to deviate. For the sake of those who haven't experienced both the anime and the manga, spoilers will be kept to a minimum.
10 Yu-Gi-Oh!
Fans of Yu-Gi-Oh! are no doubt familiar with the franchise's origins, but those less familiar with it may not be. Yu-Gi-Oh! was not initially intended to be about the card game, but rather about games in general, with the cards being just one of many. However, the cards became extremely popular, and the series shifted focus to that specifically. When it came time to adapt Yu-Gi-Oh! as an anime, the early material that didn't relate to the card game was just left out entirely, removed from continuity. This material was actually adapted before, though, in the so-called "season 0" of Yu-Gi-Oh!, but it's not canon to the rest of the series and has never been officially translated and released in the West. Since the original Yu-Gi-Oh! Duel Monsters anime, though, most Yu-Gi-Oh! series and spinoffs have been original productions rather than adaptations, even though manga versions of those series do exist.
9 Soul Eater
Soul Eater is a perfect case study of what happens when an anime catches up to the manga and runs out of material. The series is relatively faithful to the manga up until episode 37 of its 51 episode run, when the weekly anime series caught up to the monthly manga releases. With nothing left to adapt, the production studio, Bones, had to find a way to end the story themselves. Some fans were a bit upset with the anime's ending, as they felt the heroes won a bit too easily, while others actually grew to prefer the anime's ending over the manga's eventual end, due to the way certain characters and especially character deaths were handled.
8 The Promised Neverland
The Promised Neverland anime was one of the big hits of 2019, but it ran into a very different sort of problem. The first season of The Promised Neverland was marketed as a horror anime, set in this orphanage that was secretly a human farm. However, the manga had become very different by that point, essentially turning into an adventure series once the kids broke out of the orphanage. Fans of the first season of the anime weren't looking for an adventure, though--they were looking for more horror. Season 2 of The Promised Neverland skips a lot of content and moves through the story from the manga at a rapid pace, adapting 30-ish chapters for the first season and nearly 150 for the second. Huge amounts of story went missing, and many fans argue that the second season shouldn't be watched at all.
7 Bokurano
Bokurano is a 2007 mecha anime that makes for another interesting example; according to his blog, the director of the Bokurano anime, Hiroyuki Morita, was not a fan of how the manga handled its characters or story, and opted to do his own take on the series from day one. He was fervent about this to the point of encouraging fans of the manga not to watch the anime. As such, Bokurano is much more in the "inspired by" category, although some plot points persist between versions. The manga is quite brutal, with no fear of killing off characters, and Morita's comments seem to suggest that this was one of his issues with the series.
6 Trigun
The original 1998 Trigun anime was based in part on the original Trigun manga, which ran for only 20 or so chapters before it was unceremoniously canceled when the magazine it ran in went under. Trigun managed to get picked up by another magazine as a sequel series, Trigun Maximum, which ran from 1998 to 2008. As such, there are nearly 10 years of manga content that never made it into the anime simply because it didn't exist yet. Because Maximum had only just started, barely any of it is adapted into the series; modern fans might even say that over half of the original Trigun is "filler" since it doesn't come from the manga. Fortunately, in this case, Trigun Stampede is looking to be a much closer adaptation of the manga, although it took 25 years for fans to finally get that.
5 Tsubasa Reservoir Chronicle
Sometimes when the studio has to create new content for "filler" or for anime-original endings after catching up to the manga, that new content misses the mark... hard. In fact, Tsubasa Reservoir Chronicle managed to upset the series creators, manga collective CLAMP, so much that they took the rights away from the studio (Bee Train) and handed them over to Production I.G. to finish the rest of the series in OVA format. The filler content violated a core tenet of Tsubasa Reservoir Chronicle's world: that no one can ever come back from death, no matter what.
4 Boruto
Boruto began life as a film, oddly enough, and that film was then adapted into a manga and anime. Only, the manga and the anime couldn't have less to do with each other, at least at first. It might be fair to say that the Boruto anime was rushed out to capitalize on Naruto's lingering popularity after its ending, and rather than waiting for the manga to create content that could be adapted, they decided to make both the anime and manga at the same time. Since there wasn't anything to adapt, the Boruto anime created original story arcs (almost explicitly filler) to give the manga time to create a storyline that the anime would eventually pick up and begin adapting.
3 Tokyo Ghoul
Tokyo Ghoul's first season is generally considered to be a faithful adaptation of part of the series, and fans of the series were excited to see the next part adapted. However, when it came time for Tokyo Ghoul to continue, the result was Tokyo Ghoul √A, a largely anime-original second season that eventually did mix in parts of the manga, a fact which mostly just made it difficult to adapt the manga properly later. Interestingly, the new content was created by the original mangaka, putting it in an odd place in of canonicity.
2 Black Butler
Black Butler is another long-running manga that's still ongoing; since the anime was produced between 2008 and 2010, it would seem to be obvious that, like Trigun, much of the manga hasn't been adapted. However, the anime actually diverges from the manga well before it runs out of material to adapt, starting with the seventh episode of the first season, instead going off and telling its own story, changing many characters significantly in the process. The second season has essentially nothing in common with the manga aside from characters. Weirdly, the series then went back to adapting the manga, branching off into a third continuity where part of the original anime is canon and part isn't.
1 Fullmetal Alchemist (2003)
Fullmetal Alchemist's original 2003 anime may be the most famed example of an anime diverging from the manga, but the truth is that this was always meant to be the case. Since the manga had barely begun by that point (it started in 2001 and ran until 2010), it was obvious that the series would need to do its own thing if it were to end in a satisfactory manner. The 2003 anime adapts the manga for the first 30 episodes before going off in its own direction, creating a very different sequence of events and radically altering many characters. When it came time to make the Fullmetal Alchemist Brotherhood anime and adapt the manga properly, though, the studio opted not to animate what they had adapted the first time again, so neither series is actually a 100% faithful adaptation of the manga--an anime viewer would really need to watch both to get it all.
While diverging from the manga can be divisive among anime fans, these turns of events usually come about for a reason, and the production studio is simply doing the best with the situation they've been given. Some of these series (particularly FMA and Trigun) are still very well regarded, despite their divergences. Even if it's not quite what they expected, a true fan should still be able to appreciate the anime for what it is--at least as long as the manga's quality has remained.